Counting numerical semigroups by Frobenius number, multiplicity, and depth

In 1990, Backelin showed that the number of numerical semigroups with Frobenius number $f$ approaches $C_i\cdot 2^{f\mathrm{/}2}$ for constants $C_0$ and $C_1$ depending on the parity of $f$. In this paper, we generalize this result to semigroups of arbitrary depth by showing there are $\lfloor (q+1{)}^2\mathrm{/}4{\rfloor }^{f\mathrm{/}(2q-2)+o(f)}$ semigroups with Frobenius number $f$ and depth $q$. More generally, for fixed $q\ge 3$, we show that, given $(q-1)m<f<qm$, the number of numerical semigroups with Frobenius number $f$ and multiplicity $m$ is$${\left({\lfloor \frac{(q+2{)}^2}{4}\rfloor }^{\alpha \mathrm{/}2}{\lfloor \frac{(q+1{)}^2}{4}\rfloor }^{(1-\alpha )\mathrm{/}2}\right)}^{m+o(m)}$$ where $\alpha =f\mathrm{/}m-(q-1)$. Among other things, these results imply Backelin’s result, strengthen bounds on $C_i$, characterize the limiting distribution of multiplicity and genus with respect to Frobenius number, and resolve a recent conjecture of Singhal on the number of semigroups with fixed Frobenius number and maximal embedding dimension.